Asymptotic integration of $(1+\alpha )$-order fractional differential equations

Abstract

\noindent{\bf Abstract} We establish the long-time asymptotic formula of solutions to the $(1+\alpha )$--order fractional differential equation ${}_0^i{\mathcal{O}}_t^{1+\alpha }x+a(t)x=0$, $t>0$, under some simple restrictions on the functional coefficient $a(t)$, where ${}_0^i{\mathcal{O}}_t^{1+\alpha }$ is one of the fractional differential operators ${}_0{D}_t^{\alpha }(x^{\prime })$, $({}_0{D}_t^{\alpha }x{)}^{\prime }={}_0{D}_t^{1+\alpha }x$ and ${}_0{D}_t^{\alpha }(t{x}^{\prime }-x)$. Here, ${}_0{D}_t^{\alpha }$ designates the Riemann-Liouville derivative of order $\alpha \in (0,1)$. The asymptotic formula reads as $[a+O(1)]\cdot x_{small}+b\cdot x_{large}$ as $t\to +\infty$ for given $a$, $b\in \mathbb{R}$, where $x_{small}$ and $x_{large}$ represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation ${}_0^i{\mathcal{O}}_t^{1+\alpha }x=0$, $t>0$.

16 pages